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The Church-Rosser property in symmetric combinatory logic

Published online by Cambridge University Press:  12 March 2014

Katalin Bimbó
Katalin Bimbó*
Affiliation:
School of Informatics, Indiana University, 901 East 10th Street, Bloomington, IN 47408-3912., USA, E-mail: [email protected], URL: http://mypage.iu.edu/~kbimbo
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Abstract

Symmetic combinatory logic with the symmetric analogue of a combinatorially complete base (in the form of symmetric λ-calculus) is known to lack the Church-Rosser property. We prove a much stronger theorem that no symmetric combinatory logic that contains at least two proper symmetric combinatory has the Church-Rosser property. Although the statement of the result looks similar to an earlier one concerning dual combinatory logic, the proof is different because symmetric combinators may form redexes in both left and right associated terms. Perhaps surprisingly, we are also able to show that certain symmetric combinatory logics that include just one particular constant are not confluent. This result (beyond other differences) clearly sets apart symmetric combinatory logic from dual combinatory logic, since all dual combinatory systems with a single combinator or a single dual combinator are Church-Rosser. Lastly, we prove that a symmetric combinatory logic that contains the fixed point and the one-place identity combinator has the Church-Rosser property.

Keywords

symmetric combinatory logicChurch-Rosser propertydual combinatory logiccombinatory logictype theorysubstructural logicsstructurally free logicsymmetric λ-calculus
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 2 , June 2005 , pp. 536 - 556
Copyright
Copyright © Association for Symbolic Logic 2005

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